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Parabolic Boundary Harnack Inequalities with Right-Hand Side

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We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing, for the first time, a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side $f \in L^q$ for $q>n+2$. In the case of the heat equation, we also show the optimal $C^{1-\varepsilon}$ regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are $C^{1, \alpha}$ in the parabolic obstacle problem and in the parabolic Signorini problem.

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Archive for Rational Mechanics and Analysis, 2024, vol. 248, num.5

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Articles publicats en revistes (Matemàtiques i Informàtica)

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Parabolic Boundary Harnack Inequalities with Right-Hand Side